Download Projective Measurements

Slides for
Quantum Computing and Communications – An
Engineering Approach
Chapter 3
Measurements
Sándor Imre
Ferenc Balázs
Motto
"There are two possible outcomes: If the result
confirms the hypothesis, then you’ve made a
measurement. If the result is contrary to the
hypothesis, then you’ve made a discovery.“
Enrico Fermi
General Measurements
"WHY must I treat the measuring device classically?? What will
happen to me if I don’t??"
Eugene Wigner
3rd Postulate using ket notations
• Measurement statistic
• Post measurement state
• Completeness relation
Projective Measurements
(Neumann Measurements)
“Projective geometry has opened up for us with the greatest
facility new territories in our science, and has rightly been
called the royal road to our particular field of knowledge.”
Felix Klein
Measurement operators and the 3rd Postulate
in case of projective measurements
• Set of two orthogonal states
• To find
we need to solve
• We are looking in the form of
• Similarly
Measurement operators and the 3rd Postulate
in case of projective measurements
• Checking the Completeness relation
• Practical notation
• Conclusion
Measurement operators and the 3rd Postulate
in case of projective measurements
•
belong to a special set of operators called
projectors
• Properties
Measurement operators and the 3rd Postulate
in case of projective measurements
• 3rd Postulate with projectors
Measurement operators and the 3rd Postulate
in case of projective measurements
• Direct construction approach
• Indirect construction approach
Measurement using the computational
basis states
• Let us check what we have learned by means of a
simple example
• Basis vectors
and
• Measurement statistic
Measurement using the computational
basis states
• Post measurements states
• Remark: Orthogonal states can always be distinguished via
constructing appropriate measurement operators (projectors).
This is another explanation why orthogonal (classical) states can
be copied as was stated in Section 2.7 because in possession of
the exact information about such states we can build a quantum
circuit producing them.
Observable and projective
measurements
• Any observable can be represented by means of a
Hermitian operator whose eigenvalues refer to the
possible values of that observable
• Expected value of such an observable can be
calculated in an easy way
Repeated projective measurement
• What happens when we repeat a projective
measurement on the same qreqister?
• Post measurement state after the first measurement
• Post measurement state after the second
measurement
CHSH inequality with entangled
particles
• We return to Bell inequalities and investigate CHSH
inequality in the nano-world. We replace books with
entangled pairs
• Alice and Bob check the following observables
CHSH inequality with entangled
particles
CHSH inequality with entangled
particles
• Finally we obtain
which is obviously
greater than the classical result 2.
• Experiments shore up the quantum model instead of
the classical one!
• Hidden variables do not exist!
Positive Operator Valued Measurements
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Motivations to use POVM
• In certain cases either we are not interested in the
post measurement state or we are not able at all to
get it (e.g. a photon hits the detector).
• Important properties of
POVM and the 3rd Postulate
• Measurement statistic
• Post measurement state
Unknown and/or indifferent!
• Completeness relation
Can non-orthogonal states be
distinguished?
• The answer is definitely NOT because
• Remark: No set of measurement operators exist which is able to
distinguish non-orthogonal states unambiguously. This is another
explanation why non-orthogonal states can not be copied as was
stated earlier, because of lack of exact information about such
states we can not build a quantum circuit to produce them.
POVM construction example
• Our two non-orthogonal states are:
• Based on the indirect construction approach we try to
ensure that
• To achieve this we need
POVM construction example
• To fulfil the completeness relation
• Completeness relation OK since
D2 must be positive semi-definite
•
is very promising, but unfortunately wrong!
How to apply POVM operators - 0 and
1 has the same importance
• It requires
How to apply POVM operators minimising uncertainty
• Goal: to minimise
• Assuming
How to apply POVM operators minimising uncertainty
Copyright © 2005 John Wiley & Sons Ltd.
How to apply POVM operators - false
alarm vs. not happen alarm
• It is assumed that detecting 1 correctly is much more
important
Copyright © 2005 John Wiley & Sons Ltd.
Generalisation of POVM
Relations among the measurement types
• Projective measurement can be regarded as a special
POVM.
• Clearly speaking POVM is a generalised measurement
without the interest of post-measurement state +
construction rules.
• Neumark’s extension: any generalised measurement
can be implemented by means of a projective
measurement + auxiliary qbits + unitary transform.
Relations among the measurement types
Copyright © 2005 John Wiley & Sons Ltd.
Quantum computing-based solution of
the game with marbles
"Victorious warriors win first and then go to war, while defeated warriors go
to war first and then seek to win.” Sun Tzu
• We exploit entanglement and the orthogonality
between Bell states
Quantum computing-based solution of
the game with marbles
Copyright © 2005 John Wiley & Sons Ltd.